Related papers: Fractals and the two dimensional Jacobian Conjectu…

Jacobian conjecture states that if $F:\ \mathbb C^n(\mathbb R^n)\rightarrow \mathbb C^n(\mathbb R^n)$ is a polynomial map such that the Jacobian of $F$ is a nonzero constant, then $F$ is injective. This conjecture is still open for all…

The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of…

We present several versions of the Jacobian Conjecture in positive characteristic each of which if true would imply the Jacobian conjecture in characteristic 0. We test these characteristic p versions of the conjecture against several…

Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism that has an invertible Jacobian. We bring two ideas concerning the Jacobian Conjecture: First, we conjecture that for all $n$, the…

It is shown that fractal dimension can be estimated seeking a solution of functional equation defined for areas of coverages of different scales. The method proposed is compared with widely known way to estimate fractal dimension via linear…

Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then…

In this paper we consider an alternative orthogonal decomposition of the space $L^2$ associated to the $d$-dimensional Jacobi measure and obtain an analogous result to P.A. Meyer's Multipliers Theorem for d-dimensional Jacobi expansions.…

This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with…

Assuming a fractal distribution of matter in the universe, consequences that follow from the General Theory of Relativity and the Copernican Principle for fractal cosmology are examined. The change in perspective necessary to deal with a…

The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of…

Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its…

We show that the fractonic dipole-conserving algebra can be obtained as an Aristotelian (and pseudo-Carrollian) contraction of the Poincar\'e algebra in one dimension higher. Such contraction allows to obtain fracton electrodynamics from a…

In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.

We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.

In this paper, we give a survey of a geometrical theory of Jacobi forms of higher degree. And we present some geometric results and discuss some geometric problems to be investigated in the future.

A sequence is a fractal sequence if it contains itself as a proper subsequence. (The self-containment property resembles that of visual fractals) A doubly fractal sequence of integers is defined by operations called upper trimming and lower…

This paper discusses some topics of enquiry concerning fractals, functions on them, and so on.

The Image Conjecture was formulated by the third author, who showed that it implied his Vanishing Conjecture, which is equivalent to the famous Jacobian Conjecture. We prove various cases of the Image Conjecture and show how it leads to…